Local Noether Lattices Research Articles

Representations of complete regular local noether lattices

In this paper, we prove the uniqueness of the the ring of representation for any complete regular local Noether lattice. We investigate conditions for representability and obtain a form of necessary and sufficient condition.

Generalized symmetric elements generated by a prime sequence

Let \(A_1, A_2, \dots, A_n\) be a prime sequence in a local Noether lattice L. For \(k \ge 1, {\cal P} {_k}b \) denotes the set of finite joins in L of power products of the generalized symmetric elements of order k (majorization elements) in \(A_1, A_2, \dots, A_n\) together with 0 and I. We have previously showed that for \(k = 1, {\cal P} {_k}b \) is a Noetherian distributive \(\pi\)-domain. For \(n \le 3\) and for any \(k,{\cal P} {_k}b\) is again such a sub-\(\pi\)-domain of \({\cal P} {_1}b\). For \(n \ge 4\) and \(k \ge 2, {\cal P} {_k}b\) is not closed under the meet of \( {\cal P} {_1}b\). However \( {\cal P} {_k}b\) with its induced meet is again a Noetherian distributive \(\pi\)-domain. Each finite set of majorization elements asymptotically forms a distributive sublattice of \({\cal P}{_k}b\) for k sufficiently large.

P‐systems in local Noether lattices

In this paper we introduce the concept of a p‐system in a local Noether lattice and obtain several characterizations of these elements. We first obtain a topological characterization and then a characterization in terms of the existence of a certain type of decreasing sequence of elements. In addition, p‐systems are characterized in quotient lattices and completions.

Tidy Noether lattices

A Noether lattice ℒ satisfying the union condition on primes which is not a domain and in which every nonzero principal element is integrally closed is characterized in terms of its direct summands. It is shown that either: (1) ifℒ has no proper nonzero direct summands, then every nonzero principal element ofℒ is integrally closed if and only if ℒ is a local Noether lattice whose maximal element is principal and has square zero; or (2) ifℒ has a proper nonzero direct summand, then every nonzero principal element ofℒ is integrally closed if and only if for each minimal direct summandA ofℒ, the quotient lattice [0,A] is an integrally closed domain.

Noether Lattices Representable as Quotients of the Lattice of Monomially Generated Ideals of Polynomial Rings

Noether lattices were introduced by R. P. Dilworth in [5] and constitute a natural abstraction of the lattice of ideals of a Noetherian ring. In his definitive work, Dilworth showed that a minimal prime of an element generated by n principal elements has rank ≦ n. Following standard ring theoretical terminology, a local Noether lattice with (unique) maximal element M is said to be regular if M has rank n and can be generated by n principal elements.In [3], K. P. Bogart showed that a distributive regular local Noether lattice of Krull dimension n is isomorphic to RLn, the sublattice of all ideals generated by monomials of any polynomial ring (k a field). In a later paper [4], Bogart extended his results on distributive regular local Noether lattices by showing that any distributive local Noether lattice is the image of a multiplicative map θ which preserves joins, and can in fact be thought of as the related congruence lattice.

Unique factorization and small regular local noether lattices

Noether lattices were introduced by Dilworth [5]. Simply stated, a Noether lattice is a complete, modular, commutative multiplicative lattice satisfying the ascending chain condition in which every element is the join of principal elements. The resultant setting not only admits natural formulations of the fundamental definitions and results of classical ideal theory, but is simultaneously rich enough to yield, for example, the Noether decomposition theorems, the Intersection Theorem, and the Principal Ideal Theorem [Sj. The key to the richness of the setting lies, of course, in the definition of a principal element. An element E is said to be principal if it satisfies the two identities BE A A = (B A (A : E))E and (B v AE) : E = (B : E) v A, for all A and B. These identities are easily verified for the principal ideals of a commutative ring. In [4], Bogart obtained the surprising result that a regular local Noether lattice with exactly n < 3 rank 1 primes is isomorphic to RL, , the Noether sublattice of k[x, ,..., x,] (k a field) consisting of all ideals which are generated by monomials in x1 , xs ,..., x, . He conjectured the validity of the result for all n, but noted that, due to the heavy dependence of the case n = 3 on the observation that in a regular local Noether lattice of dimension 2, all rank 1 primes are principal, the general case might be closely tied to a generalization of the Auslander-Buchsbaum theorem that regular local rings are UFDs. In [2], Anderson conjectured that any local Noether lattice with exactly n rank 1 primes which is a n-domain (i.e., every principal element is a product of primes, or, equivalently in this case, every rank 1 prime is principal) is isomorphic to RL, , and therefore regular. In this paper we establish the validity of both conjectures. While our results do not yield a generalization of the Auslander-Buchsbaum theorem as hoped, they do at least show that any counterexample to the generalization will involve an infinite number of primes.

A note on regular local Noether lattices II

Let (R, M) be a local ring and let R* be the M-adic ring completion of R. It is well known that R is a regular local ring if and only if R* is a regular local ring. The purpose of the note is to show that this result is essentially a consequence of a more general theory concerning local Noether lattices which was developed in [6].

Weak join-principal element lattices

In this paper we study local Noether lattices in which the maximal element is join-principal or weak join-principal. A characterization of maximal join-principal elements involving distributivity and independence is given and the distributive local Noether lattices with weak join-principal maximal elements are determined.

Join-principal element lattices

Let ( L , M ) (\mathfrak {L},M) be a local Noether lattice. If the maximal element M M is meet principal, it is well known and easily seen that every element of L \mathfrak {L} is meet principal. In this note, we obtain the corresponding result for M M join-principal. We also consider join-principal elements generally under the assumption of the weak union condition and show, for example, that the square of a join-principal element is principal.

Regularity in Terms of Reductions in Local Noether Lattices

Regularity in Terms of Reductions in Local Noether Lattices

Regularity in terms of reductions in local Noether lattices

An $(n,d)$-sequence in a local Noether lattice $(L,M)$ is a sequence of words which satisfy certain factorization properties. If $(L,M)$ satisfies the union condition, there exist $(n,d)$-sequences which can be extended to minimal bases for the powers of $M$. Consequently, if $(L,M)$ satisfies the union condition, $(L,M)$ is regular if and only if $M$ is a minimal reduction.

Prime sequences and distributivity in local Noether lattices

Prime sequences and distributivity in local Noether lattices

Small Regular Local Noether Lattices. I

In a recent paper the author has discussed the structure of regular local Noether lattices. In this paper it is proved that if a regular local Noether lattice has precisely $3$ minimal primes, then it is isomorphic to $R{L_3}$, the lattice of the lattice of ideals of $F[{x_1},{x_2},{x_3}]$ generated by the principal ideals $({x_1}),({x_2})$ and $({x_3})$ under join and multiplication.

M-Primary Elements of a Local Noether Lattice

In this paper, we consider the extent to which a local Noether lattice (ℒ, M) is characterized by the sub-multiplicative lattice, denoted δℒ, of M-primary elements. (Here we use the notation (ℒ, M) to indicate that M is the maximal element of ℒ.) In particular, we call ℒM-complete if, given any decreasing sequence {Ai} of elements and any n ≧ 1, it follows that Ai ≦ A V Mn for large i, where A = ΛAi And we show that, given two Mi-complete local Noether lattices (ℒ1, M1) and (ℒ2, M2), with δℒ1 ≅ δℒ2, it follows that ℒ1 ≅ ℒ2. Further, we show that any local Noether lattice (ℒ, M) is a sublattice of a local Noether lattice (ℒ*, M) which is M-complete and such that δℒ = δℒ*.

Small regular local Noether lattices. I

In a recent paper the author has discussed the structure of regular local Noether lattices. In this paper it is proved that if a regular local Noether lattice has precisely $3$ minimal primes, then it is isomorphic to $R{L_3}$, the lattice of the lattice of ideals of $F[{x_1},{x_2},{x_3}]$ generated by the principal ideals $({x_1}),({x_2})$ and $({x_3})$ under join and multiplication.

Structure theorems for regular local Noether lattices.

Structure theorems for regular local Noether lattices.